Question
Horizontal lines e and f are intersected by lines a and b. At the intersection of lines a and e, the uppercase left angle is 75 degrees. At the intersection of lines b and e, the uppercase right angle is 115 degrees. At the intersection of lines a and f, the bottom right angle is 75 degrees.
Which lines are parallel? Justify your answer.
Lines a and b are parallel because their alternate exterior angles are congruent.
Lines a and b are parallel because their same side exterior angles are supplementary.
Lines e and f are parallel because their alternate exterior angles are congruent.
Lines e and f are parallel because their same side exterior angles are congruent.
Which lines are parallel? Justify your answer.
Lines a and b are parallel because their alternate exterior angles are congruent.
Lines a and b are parallel because their same side exterior angles are supplementary.
Lines e and f are parallel because their alternate exterior angles are congruent.
Lines e and f are parallel because their same side exterior angles are congruent.
Answers
GPT-4o mini
To determine which lines are parallel based on the angles given, let's analyze the intersections and the properties of corresponding and alternate angles.
1. **Angles at the Intersection of Lines a and e:**
- The uppercase left angle is 75 degrees. This angle is an alternate interior angle.
- Therefore, the angle adjacent to it, which we call the lowercase right angle, is supplementary to 75 degrees. Hence, this angle is 105 degrees (180 - 75 = 105).
2. **Angles at the Intersection of Lines b and e:**
- The uppercase right angle at this intersection is 115 degrees.
3. **Angles at the Intersection of Lines a and f:**
- The bottom right angle is 75 degrees.
### Step 1: Analyze lines a and b
- At the intersection with line e, line a has an angle measuring 75 degrees.
- At the intersection with line e for line b, there is an angle measuring 115 degrees.
- If we analyze these angles:
- The 75-degree angle (line a) and the 115-degree angle (line b) are adjacent interior angles on the same side of line e.
- They add up to 190 degrees (75 + 115), which is not supplementary. Thus, lines a and b cannot be parallel based on this configuration.
### Step 2: Analyze lines e and f
- At the intersection with line e, line a has an angle of 75 degrees.
- At the intersection with line f, there is a 75-degree angle from line a which corresponds to the angle at line b.
- Angles related to lines e and f:
- The angle from line a (75 degrees) and the corresponding angle to line b (which is opposite, and should also be 75 degrees in the same transversal view) implies the angles are congruent.
- Line e's angles with respect to line a and line f reflect this congruence—the alternate and corresponding angles from transversal lines (a and b) to lines e and f indicate consistency.
### Conclusion:
- Lines **e and f are parallel** because their alternate interior angles with respect to transversal lines (a and b) are congruent (75 degrees at their respective intersections). Thus, their alternate exterior angles are equal and confirm that lines e and f are parallel.
- Therefore, the correct justification is: **"Lines e and f are parallel because their alternate exterior angles are congruent."**
1. **Angles at the Intersection of Lines a and e:**
- The uppercase left angle is 75 degrees. This angle is an alternate interior angle.
- Therefore, the angle adjacent to it, which we call the lowercase right angle, is supplementary to 75 degrees. Hence, this angle is 105 degrees (180 - 75 = 105).
2. **Angles at the Intersection of Lines b and e:**
- The uppercase right angle at this intersection is 115 degrees.
3. **Angles at the Intersection of Lines a and f:**
- The bottom right angle is 75 degrees.
### Step 1: Analyze lines a and b
- At the intersection with line e, line a has an angle measuring 75 degrees.
- At the intersection with line e for line b, there is an angle measuring 115 degrees.
- If we analyze these angles:
- The 75-degree angle (line a) and the 115-degree angle (line b) are adjacent interior angles on the same side of line e.
- They add up to 190 degrees (75 + 115), which is not supplementary. Thus, lines a and b cannot be parallel based on this configuration.
### Step 2: Analyze lines e and f
- At the intersection with line e, line a has an angle of 75 degrees.
- At the intersection with line f, there is a 75-degree angle from line a which corresponds to the angle at line b.
- Angles related to lines e and f:
- The angle from line a (75 degrees) and the corresponding angle to line b (which is opposite, and should also be 75 degrees in the same transversal view) implies the angles are congruent.
- Line e's angles with respect to line a and line f reflect this congruence—the alternate and corresponding angles from transversal lines (a and b) to lines e and f indicate consistency.
### Conclusion:
- Lines **e and f are parallel** because their alternate interior angles with respect to transversal lines (a and b) are congruent (75 degrees at their respective intersections). Thus, their alternate exterior angles are equal and confirm that lines e and f are parallel.
- Therefore, the correct justification is: **"Lines e and f are parallel because their alternate exterior angles are congruent."**